Transverse vibration of Bernoulli–Euler beams carrying point masses and taking into account their rotatory inertia: Exact solution

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Journal of Sound and Vibration 303 (2007) 895–908

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Short Communication

Transverse vibration of Bernoulli–Euler beams carryingpoint masses and taking into account their rotatory inertia:

Exact solution

Santiago Maiza,b, Diana V. Bambilla,b,�, Carlos A. Rossita,b, P.A.A. Lauraa

aDepartamento de Ingenierıa, Instituto de Mecanica Aplicada, Universidad Nacional del Sur,

Av. Alem 1253 (B8000CPB), Bahıa Blanca, ArgentinabConsejo Nacional de Investigaciones Cientıficas y Tecnicas (CONICET), Argentina

Received 20 October 2005; received in revised form 30 May 2006; accepted 20 December 2006

Available online 26 March 2007

Abstract

The situation of structural elements supporting motors or engines attached to them is usual in technological

applications. The operation of the machine may introduce severe dynamic stresses on the beam. It is important, then, to

know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural

elements. An exact solution for the title problem is obtained in closed-form fashion, considering general boundary

conditions by means of translational and rotatory springs at both ends. The model allows to analyze the influence of the

masses and their rotatory inertia on the dynamic behavior of beams with all the classic boundary conditions, and also, as

particular cases, to determine the frequencies of continuous beams.

r 2007 Elsevier Ltd. All rights reserved.

1. Introduction

Extensive research has been carried out with regard to the vibration analysis of beams carrying concentratedmasses at arbitrary positions and additional complexities.

Chen [1] solved analytically the problem of a vibrating simply supported beam carrying a concentrated massat its center, introducing the mass by the Dirac delta function. Laura, Pombo and Susemihl [2], Laura,Maurizi and Pombo [3] studied the cantilever beam carrying a lumped mass at the top, obtaining analyticalsolution, introducing the mass in the boundary conditions. Dowell [4] in a thorough paper studied generalproperties of beams carrying springs and concentrated masses, making useful observations over the matter.Laura, Verniere de Irassar and Ficcadenti [5] used Rayleigh–Ritz method to study continuous beams subjectedto axial forces and carrying concentrated masses. Gurgoze [6,7] used the ‘‘normal mode summation’’technique to determine the fundamental frequency of cantilever beams carrying masses and torsional springs.Liu, Wu and Huang [8] used the Laplace transformation technique to formulate the frequency equation for

ee front matter r 2007 Elsevier Ltd. All rights reserved.

v.2006.12.028

ing author. Tel.: +54291 4595100; fax: +54 291 4595157.

ess: [email protected] (D.V. Bambill).

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ARTICLE IN PRESSS. Maiz et al. / Journal of Sound and Vibration 303 (2007) 895–908896

beams carrying intermediate concentrated masses. Other studies of the influence of these factors on slenderbeam vibrations are given in Refs. [9–17].

In most of the studies mentioned above the influence of the rotatory inertia of the attached mass is not takeninto account. Laura, Filipich and Cortınez [18] considered the rotatory inertia of concentrated massesattached to beams and plates, obtaining fundamental frequencies of coupled systems by means of theRayleigh–Ritz and Dunkerley methods. Chang [19] studied a simply supported Rayleigh beam carrying arigidly attached centered mass. He determined the natural frequencies and normal modes of the system but hekept the position of the mass fixed.

In the present paper, we describe the determination of the natural frequencies of vibration of aBernoulli–Euler beam with general boundary conditions at the ends, carrying a finite number of masses atarbitrary positions, having into account their rotatory inertia. The generality of this approach is based onusing translational and rotational springs at both ends, which allow us to represent all the possiblecombinations of classical boundary conditions, as well as elastic restraints. Therefore, the purpose of thisstudy is to present a general solution of the problem and tabulate the first five frequencies for a wide range ofsystem parameters which may help in comparisons of approximate methods.

The model may also be used, as it is known [20], in the problem of whirling of a rotating shaft of uniformcross-section, where the masses model transmission elements or inertia wheels, since the same generalequations describe both systems.

2. Mathematical procedure

As shown in Fig. 1, the model considered is a beam with concentrated masses m1 and m2 located at x1 andx2, respectively, where x is the spatial coordinate along the beam of length l. I1 ¼ m1 r1

2; I2 ¼ m2 r22 are the

moments of inertia of the attached masses, where r1 and r2 are their radii of gyration with respect to theneutral axis of the beam; k1 and k2 are the translational stiffness while k3 and k4 are rotational stiffness.

In order to find the natural frequencies of the system one assumes that the beam deflection v(x, t) may beexpressed in the form

v x; tð Þ ¼ V xð Þ cos ot, (1)

where o is the natural circular frequency.Taking into account Eq. (1), the problem under consideration is governed by the following differential

equation:

d4V

dx4� b4V ¼ 0, (2)

where b4 ¼ ðAr=EIÞo2, r is the beam material density, A is the cross-sectional area, I is the cross-sectionalmoment of inertia and E is the Young’s modulus.

Fig. 1. Bernoulli–Euler beam elastically supported carrying two masses.

ARTICLE IN PRESSS. Maiz et al. / Journal of Sound and Vibration 303 (2007) 895–908 897

The general solution of the differential Eq. (2) may be written as a piecewise function:

V 1 xð Þ ¼ C1 cosh bxþ C2 sinh bxþ C3 cos bxþ C4 sin bx for 0pxpx1,

V 2 xð Þ ¼ C5 cosh bxþ C6 sinh bxþ C7 cos bxþ C8 sin bx for x1pxpx2,

V 3 xð Þ ¼ C9 cosh bxþ C10 sinh bxþ C11 cos bxþ C12 sin bx for x2pxpl, ð3Þ

where Ci are constants to be determined with the boundary conditions at x ¼ 0 and l and the continuityequations at x1 and x2 while V1, V2 and V3 are, respectively, the left, central and right transverse displacementsdivided at the points where the concentrated masses are attached.

Introducing the following non-dimensional coordinates:

Z ¼x

lone has Z1 ¼

x1

land Z2 ¼

x2

l, (4)

the boundary conditions are:

d3V 1 Zð ÞdZ3

��Z¼0¼ �K1V 1 Zð Þ

��Z¼0, (5)

Table 1

First five eigenvalues (bnl) for symmetric location of masses on a clamped–clamped beam

Z1 ¼ 0.25; Z2 ¼ 0.75 bnl c1 ¼ c2 ¼ 0 c1 ¼ c2 ¼ 0.01 c1 ¼ c2 ¼ 0.05 c1 ¼ c2 ¼ 0.1

M1 ¼M2 ¼ 0 b1l 4.7300

b2l 7.8532

b3l 10.9956

b4l 14.1372

b5l 17.2788

M1 ¼M2 ¼ 0.01 b1l 4.7126 4.7125 4.7112 4.7071

b2l 7.7732 7.7731 7.7723 7.7696

b3l 10.8958 10.8956 10.8899 10.8714

b4l 14.1150 14.1125 14.0520 13.8602

b5l 17.2557 17.2513 17.1426 16.7908

M1 ¼M2 ¼ 0.1 b1l 4.5668 4.5663 4.5554 4.5217

b2l 7.1911 7.1908 7.1855 7.1671

b3l 10.2346 10.2325 10.1796 9.9795

b4l 13.9713 13.9472 13.3525 11.7542

b5l 17.1148 17.0715 15.9720 13.5895

M1 ¼M2 ¼ 0.5 b1l 4.0973 4.0961 4.0663 3.9755

b2l 5.8984 5.8980 5.8893 5.8555

b3l 9.1453 9.1356 8.8716 7.9804

b4l 13.7527 13.6401 11.2437 8.5500

b5l 16.9258 16.7178 12.9941 10.8372

M1 ¼M2 ¼ 1 b1l 3.7335 3.7320 3.6959 3.5868

b2l 5.1746 5.1743 5.1656 5.1306

b3l 8.7418 8.7220 8.1800 6.9010

b4l 13.6791 13.4578 9.8682 7.2687

b5l 16.8681 16.4514 11.6278 10.2256

M1 ¼M2 ¼ 2 b1l 3.3053 3.3037 3.2659 3.1514

b2l 4.4574 4.4571 4.4491 4.4160

b3l 8.4667 8.4261 7.3841 5.8827

b4l 13.6312 13.1901 8.4819 6.1460

b5l 16.8320 15.9869 10.6332 9.8684


d2V 1 Zð ÞdZ2

��Z¼0¼ K3

dV1 Zð ÞdZ

��Z¼0

, (6)

d3V 3 Zð ÞdZ3

��Z¼1¼ K2V 3 Zð Þ

��Z¼1, (7)

d2V3 Zð ÞdZ2

��Z¼1¼ �K4

dV3 Zð ÞdZ

��Z¼1

. (8)

The continuity equations of the beam at the position Z1 are:

V 1 Zð Þ��Z¼Z1¼ V2 Zð Þ

��Z¼Z1

, (9)

dV 1 Zð ÞdZ

��Z¼Z1

¼dV2 Zð ÞdZ

��Z¼Z1

, (10)

d3V1 Zð ÞdZ3

��Z¼Z1

þM1 blð Þ4V1 Zð Þ��Z¼Z1¼

d3V2 Zð ÞdZ3

��Z¼Z1

, (11)

d2V 1 Zð ÞdZ2

��Z¼Z1

�M1c21 blð Þ4

dV1 Zð ÞdZ

��Z¼Z1

¼d2V2 Zð ÞdZ2

��Z¼Z1

(12)

Table 2

First five eigenvalues (bnl) for asymmetric location of masses on a clamped–clamped beam

Z1 ¼ 0.25; Z2 ¼ 0.5 bnl c1 ¼ c2 ¼ 0 c1 ¼ c2 ¼ 0.01 c1 ¼ c2 ¼ 0.05 c1 ¼ c2 ¼ 0.1

M1 ¼M2 ¼ 0.01 b1l 4.6921 4.6921 4.6915 4.6895

b2l 7.8128 7.8125 7.8061 7.7861

b3l 10.8932 10.8931 10.8903 10.8810

b4l 14.1262 14.1236 14.0593 13.8577

b5l 17.1859 17.1836 17.1287 16.9486

M1 ¼M2 ¼ 0.1 b1l 4.4053 4.4051 4.4003 4.3856

b2l 7.4860 7.4841 7.4361 7.2818

b3l 10.2227 10.2217 10.1940 10.0654

b4l 14.0604 14.0336 13.3904 11.9149

b5l 16.6703 16.6471 16.0376 13.7828

M1 ¼M2 ¼ 0.5 b1l 3.7027 3.7022 3.6922 3.6606

b2l 6.4814 6.4778 6.3855 6.0575

b3l 9.2683 9.2606 9.0218 8.0269

b4l 13.9693 13.8313 11.3901 9.4410

b5l 16.0876 15.9755 12.9703 10.2982

M1 ¼M2 ¼ 1 b1l 3.2772 3.2768 3.2658 3.2314

b2l 5.7693 5.7658 5.6755 5.3312

b3l 9.0003 8.9827 8.3750 6.9111

b4l 13.9388 13.6573 10.2652 8.0475

b5l 15.9243 15.7069 11.3195 9.8784

M1 ¼M2 ¼ 2 b1l 2.8399 2.8394 2.8287 2.7949

b2l 5.0077 5.0046 4.9240 4.5992

b3l 8.8463 8.8084 7.5086 5.8790

b4l 13.9185 13.3490 9.0757 6.8012

b5l 15.8239 15.3957 10.2276 9.6737


and at Z2

V 2 Zð Þ��Z¼Z2¼ V3 Zð Þ

��Z¼Z2

, (13)

dV 2 Zð ÞdZ

��Z¼Z2

¼dV 3 Zð ÞdZ

��Z¼Z2

, (14)

d3V2 Zð ÞdZ3

��Z¼Z2

þM2 blð Þ4V2 Zð Þ��Z¼Z2¼

d3V3 Zð ÞdZ3

��Z¼Z2

, (15)

d2V2 Zð ÞdZ2

��Z¼Z2

�M2c22 blð Þ4dV 2 Zð ÞdZ

��Z¼Z2

¼d2V 3 Zð ÞdZ2

��Z¼Z2

, (16)

Table 3

First five eigenvalues (bnl) for symmetric location of masses on a simply supported beam

Z1 ¼ 0.25; Z2 ¼ 0.75 bnl c1 ¼ c2 ¼ 0 c1 ¼ c2 ¼ 0.01 c1 ¼ c2 ¼ 0.05 c1 ¼ c2 ¼ 0.1

M1 ¼M2 ¼ 0 b1l 3.1416

b2l 6.2832

b3l 9.4248

b4l 12.5664

b5l 15.7080

M1 ¼M2 ¼ 0.01 b1l 3.1261 3.1261 3.1257 3.1246

b2l 6.2218 6.2218 6.2218 6.2218

b3l 9.3790 9.3786 9.3687 9.3376

b4l 12.5664 12.5644 12.5167 12.3679

b5l 15.6328 15.6309 15.5845 15.4321

M1 ¼M2 ¼ 0.1 b1l 3.0013 3.0012 2.9983 2.9892

b2l 5.7745 5.7745 5.7745 5.7745

b3l 9.0595 9.0559 8.9674 8.6820

b4l 12.5664 12.5465 12.0741 10.8225

b5l 15.1713 15.1541 14.6979 13.3007

M1 ¼M2 ¼ 0.5 b1l 2.6393 2.6390 2.6315 2.6085

b2l 4.7664 4.7664 4.7664 4.7664

b3l 8.4744 8.4594 8.0892 7.1123

b4l 12.5664 12.4671 10.4963 8.0784

b5l 14.5617 14.4846 12.5720 10.8300

M1 ¼M2 ¼ 1 b1l 2.3832 2.3828 2.3740 2.3469

b2l 4.1920 4.1920 4.1920 4.1920

b3l 8.2394 8.2114 7.5328 6.2114

b4l 12.5664 12.3679 9.3276 6.8955

b5l 14.3802 14.2279 11.4423 10.2253

M1 ¼M2 ¼ 2 b1l 2.0960 2.0956 2.0864 2.0583

b2l 3.6171 3.6171 3.6171 3.6171

b3l 8.0730 8.0190 6.8399 5.3282

b4l 12.5664 12.1712 8.0784 5.8419

b5l 14.2680 13.9592 10.5691 9.8684


where

K1 ¼k1l

3

EI; K2 ¼

k2l3

EI; K3 ¼

k3l

EI; K4 ¼

k4l

EI,

Mi ¼mi

rAl,

ci ¼ri

l. ð17Þ

3. Frequency equation

Substituting Eq. (3) into Eqs. (5)–(16), taking into account Eq. (4) one obtains, after appropriate non-dimensionalization, the following system of equations expressed as

a1�1 a1�2 � � � a1�12

a2�1 a2�2 � � � a2�12

� � � � � �

� � � � � �

� � � � � �

a12�1 a12�2 � � � a12�12

26666666664

37777777775

C1

C2

�

�

�

C12

26666666664

37777777775

¼

0

0

�

�

�

0

2666666664

3777777775. (18)

Table 4

First five eigenvalues (bnl) for asymmetric location of masses on a simply supported beam

Z1 ¼ 0.25; Z2 ¼ 0.5 bnl c1 ¼ c2 ¼ 0 c1 ¼ c2 ¼ 0.01 c1 ¼ c2 ¼ 0.05 c1 ¼ c2 ¼ 0.1

M1 ¼M2 ¼ 0.01 b1l 3.1185 3.1184 3.1183 3.1177

b2l 6.2524 6.2523 6.2494 6.2403

b3l 9.3558 9.3556 9.3509 9.3356

b4l 12.5664 12.5644 12.5168 12.3684

b5l 15.5961 15.5951 15.5714 15.4950

M1 ¼M2 ¼ 0.1 b1l 2.9415 2.9414 2.9401 2.9359

b2l 6.0161 6.0151 5.9914 5.9175

b3l 8.8650 8.8637 8.8302 8.6981

b4l 12.5664 12.5465 12.0735 10.8986

b5l 14.9527 14.9422 14.6718 13.4418

M1 ¼M2 ¼ 0.5 b1l 2.4946 2.4945 2.4916 2.4824

b2l 5.3428 5.3403 5.2788 5.0881

b3l 7.9643 7.9604 7.8441 7.2183

b4l 12.5664 12.4664 10.5152 8.8187

b5l 14.2171 14.1610 12.4431 9.6262

M1 ¼M2 ¼ 1 b1l 2.2162 2.2161 2.2128 2.2027

b2l 4.8384 4.8355 4.7649 4.5445

b3l 7.6317 7.6240 7.3718 6.3101

b4l 12.5664 12.3649 9.4562 7.9257

b5l 14.0212 13.9073 10.8964 8.5582

M1 ¼M2 ¼ 2 b1l 1.9256 1.9254 1.9222 1.9121

b2l 4.2553 4.2525 4.1825 3.9609

b3l 7.4180 7.4019 6.8212 5.4070

b4l 12.5664 12.1594 8.4533 6.7602

b5l 13.9050 13.6744 9.4571 8.1863


The non-triviality condition is established by solving:

det Að Þ ¼ 0, (19)

where A is the matrix of the coefficients ai�j of the system and the roots (bnl) are the eigenvalues of theproblem. Eq. (18) is a complicated function of bnl. A numerical technique, the Newton–Raphson algorithm,was used in the present paper to find the roots.

Note that the dimension of the A matrix is (4+4n)� (4+4n) in the case of a beam carrying n concentratedmasses; therefore, if the system under study has a symmetrical configuration it will be convenient to analyzeonly half of the beam to calculate symmetric and antisymmetric vibration modes.

4. Numerical results

The first five eigenvalues were obtained for different combinations of classical boundary conditions,available in the literature. Also, some special configurations of beams are calculated as particular cases of theproposed model.

Table 5

First five eigenvalues (bnl) for symmetric location of masses on a cantilever beam

Z1 ¼ 0.25; Z2 ¼ 0.75 bnl c1 ¼ c2 ¼ 0 c1 ¼ c2 ¼ 0.01 c1 ¼ c2 ¼ 0.05 c1 ¼ c2 ¼ 0.1

M1 ¼M2 ¼ 0 b1l 1.8751

b2l 4.6941

b3l 7.8548

b4l 10.9955

b5l 14.1372

M1 ¼M2 ¼ 0.01 b1l 1.8669 1.8669 1.8668 1.8665

b2l 4.6851 4.6850 4.6827 4.6757

b3l 7.7887 7.7887 7.7869 7.7813

b4l 10.9048 10.9046 10.8999 10.8850

b5l 14.1171 14.1147 14.0569 13.8735

M1 ¼M2 ¼ 0.1 b1l 1.8003 1.8002 1.7994 1.7967

b2l 4.6083 4.6074 4.5867 4.5240

b3l 7.3191 7.3184 7.3026 7.2516

b4l 10.3067 10.3050 10.2639 10.1052

b5l 13.9865 13.9634 13.3953 11.8800

M1 ¼M2 ¼ 0.5 b1l 1.6000 1.5999 1.5976 1.5903

b2l 4.3191 4.3162 4.2466 4.0495

b3l 6.3836 6.3800 6.2961 6.0715

b4l 9.3381 9.3312 9.1379 8.2312

b5l 13.7841 13.6761 11.4015 9.2243

M1 ¼M2 ¼ 1 b1l 1.4529 1.4528 1.4499 1.4411

b2l 4.0343 4.0305 3.9408 3.6874

b3l 5.9799 5.9712 5.7797 5.3853

b4l 8.9843 8.9709 8.5646 7.0960

b5l 13.7146 13.5026 10.1527 8.5116

M1 ¼M2 ¼ 2 b1l 1.2838 1.2837 1.2806 1.2712

b2l 3.6358 3.6319 3.5381 3.2631

b3l 5.7009 5.6803 5.2724 4.6783

b4l 8.7435 8.7169 7.8393 6.0303

b5l 13.6691 13.2466 9.0712 8.0572


4.1. Clamped– clamped beam

This case is obtained by assuming the following values for the coefficients of translational and rotationalstiffness:

K1!1; K2!1; K3 !1 and K4 !1. (20)

In Table 1, the first five eigenvalues (bnl) for the problem of transverse vibration where there is a symmetriclocation of masses are shown. Results in Table 2 are for an asymmetric location of masses on the beam.

4.2. Simply supported beam

In this case the translational stiffness coefficients assume a value, which approaches infinity at both endswhile the rotational stiffness coefficients vanish:

K1!1; K2!1; K3 ¼ 0 and K4 ¼ 0. (21)

The eigenvalues are shown on Tables 3 and 4 for a symmetric and asymmetric location of masses on the beamrespectively.

4.2. Cantilever beam

In this case the end Z ¼ 0 is clamped and the end Z ¼ 1 is free, therefore

K1!1; K2 ¼ 0; K3 !1 and K4 ¼ 0. (22)

Results of eigenvalues for two different locations of the masses are shown in Tables 5 and 6.

Table 6

First five eigenvalues (bnl) for asymmetric location of masses on a cantilever beam

Z1 ¼ 0.25; Z2 ¼ 0.5 bnl c1 ¼ c2 ¼ 0 c1 ¼ c2 ¼ 0.01 c1 ¼ c2 ¼ 0.05 c1 ¼ c2 ¼ 0.1

M1 ¼M2 ¼ 0.01 b1l 1.8728 1.8728 1.8727 1.8724

b2l 4.6627 4.6626 4.6620 4.6602

b3l 7.8141 7.8138 7.8078 7.7888

b4l 10.8925 10.8924 10.8896 10.8803

b5l 14.1262 14.1235 14.0592 13.8573

M1 ¼M2 ¼ 0.1 b1l 1.8523 1.8522 1.8514 1.8490

b2l 4.4279 4.4277 4.4232 4.4090

b3l 7.4885 7.4866 7.4417 7.2971

b4l 10.2160 10.2149 10.1879 10.0621

b5l 14.0603 14.0335 13.3891 11.9090

M1 ¼M2 ¼ 0.5 b1l 1.7711 1.7709 1.7677 1.7579

b2l 3.8880 3.8875 3.8759 3.8384

b3l 6.5059 6.5026 6.4207 6.1329

b4l 9.2404 9.2331 9.0069 8.0333

b5l 13.9690 13.8307 11.3806 9.4421

M1 ¼M2 ¼ 1 b1l 1.6881 1.6879 1.6828 1.6676

b2l 3.5984 3.5977 3.5809 3.5261

b3l 5.8179 5.8151 5.7418 5.4687

b4l 8.9619 8.9453 8.3707 6.9231

b5l 13.9383 13.6562 10.2484 8.0525

M1 ¼M2 ¼ 2 b1l 1.5636 1.5633 1.5565 1.5363

b2l 3.3385 3.3374 3.3101 3.2221

b3l 5.0967 5.0946 5.0421 4.8403

b4l 8.8007 8.7651 7.5268 5.9004

b5l 13.9179 13.3470 9.0692 6.8091

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Fig. 2. Continuous beam.

Fig. 3. Symmetrical beam with four masses.

Table 7

First five eigenvalues (bnl) for two different configurations of a continuous beam

b1l b2l b3l b4l b5l

Z1 ¼ 0.25; Z2 ¼ 0.75 7.8532 12.5664 14.1372 15.7064 20.4204

Z1 ¼ 0.25; Z2 ¼ 0.5 7.1711 12.5664 13.7741 16.6419 19.8539

S. Maiz et al. / Journal of Sound and Vibration 303 (2007) 895–908 903

4.3. Continuous beam

One type of structural element that could be represented with this model is the continuous beam as shown inFig. 2. Obviously, in the case of a whirling problem, the system will correspond to a rotating shaft withintermediate bearings. In this case the following parameters must be taken:

K1!1; K2!1; K3 ¼ 0; K4 ¼ 0,

M1!1; M2 !1; c1 ¼ 0 and c2 ¼ 0. ð23Þ

The frequency coefficients are shown on Table 7 for a location of the masses (intermediate supports) atZ1 ¼ 0.25 and Z2 ¼ 0.75 and at Z1 ¼ 0.25 and Z2 ¼ 0.5.

ARTICLE IN PRESS

Fig. 4. Half-beam in a symmetric mode of vibration.

Fig. 5. Half-beam in an antisymmetric mode of vibration.

S. Maiz et al. / Journal of Sound and Vibration 303 (2007) 895–908904

4.5. Symmetric beam configuration with four masses attached

Making use of the symmetric configuration of the beam, the case shown in Fig. 3 can be solved. As it isknown, the symmetric normal modes of the system of Fig. 3 can be obtained by means of the configurationshown in Fig. 4 and the antisymmetric modes can be calculated by means of the configuration shown in Fig. 5.

The case of a clamped–clamped beam with four masses added in a symmetrical way, may be represented bythe present model, assuming the following values for the stiffness coefficients:

K1!1 and K3!1. (24)

In order to obtain the symmetric modes (see Fig. 4), one must consider

K2 ¼ 0 and K4!1 (25)

and for the antisymmetric modes, as shown in Fig. 5

K2!1 and K4 ¼ 0. (26)

The first five eigenvalues of this problem are shown on Table 8, for a symmetric arbitrary location of thefour masses.

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Table 8

First five eigenvalues (bnl) for a symmetric location of four masses on a clamped–clamped beam

Z1 ¼ 0.125; Z2 ¼ 0.375 bnl c1 ¼ c2 ¼ 0 c1 ¼ c2 ¼ 0.01 c1 ¼ c2 ¼ 0.05 c1 ¼ c2 ¼ 0.1

M1 ¼M2 ¼ 0 b1l 4.7300

b2l 7.8532

b3l 10.9956

b4l 14.1372

b5l 17.2788

M1 ¼M2 ¼ 0.01 b1l 4.6840 4.6840 4.6826 4.6782

b2l 7.7796 7.7792 7.7697 7.7399

b3l 10.9328 10.9310 10.8880 10.7556

b4l 13.8857 13.8851 13.8706 13.8239

b5l 17.0445 17.0409 16.9556 16.6877

M1 ¼M2 ¼ 0.1 b1l 4.3491 4.3487 4.3392 4.3099

b2l 7.2352 7.2325 7.1689 6.9764

b3l 10.3944 10.3819 10.0848 9.2616

b4l 12.3091 12.3056 12.2171 11.8915

b5l 15.7406 15.7063 14.9448 13.3483

M1 ¼M2 ¼ 0.5 b1l 3.5945 3.5937 3.5757 3.5210

b2l 5.9801 5.9753 5.8617 5.5285

b3l 8.7765 8.7569 8.2499 6.9649

b4l 9.6195 9.6142 9.4698 8.9339

b5l 14.3751 14.1862 11.5622 9.5382

M1 ¼M2 ¼ 1 b1l 3.1633 3.1625 3.1435 3.0865

b2l 5.2591 5.2542 5.1380 4.7995

b3l 7.7375 7.7194 7.2232 5.9649

b4l 8.3269 8.3217 8.1776 7.6522

b5l 14.0575 13.6736 9.9683 8.0941

M1 ¼M2 ¼ 2 b1l 2.7309 2.7301 2.7119 2.6577

b2l 4.5370 4.5325 4.4235 4.1072

b3l 6.6774 6.6616 6.2123 5.0638

b4l 7.1145 7.1097 6.9766 6.4975

b5l 13.8827 13.1317 8.4936 6.8383

Fig. 6. Second mode shape of a simply supported beam with two masses attached. —— Mi ¼ 0; � � � � � � Mi ¼ 0.5; – � – Mi ¼ 1;

—�— Mi ¼ 2 and ci ¼ 0.1.


ARTICLE IN PRESS

Fig. 7. Second mode shape of a simply supported beam with two masses attached. —— ci ¼ 0; � � � � � � ci ¼ 0.01; – � – ci ¼ 0.05;

—�— ci ¼ 0.1 and Mi ¼ 1.

Fig. 8. Third mode shape of a simply supported beam with two masses attached. —— Mi ¼ 0; � � � � � � Mi ¼ 0.5; – � – Mi ¼ 1;

—�— Mi ¼ 2 and ci ¼ 0.1.

S. Maiz et al. / Journal of Sound and Vibration 303 (2007) 895–908906

5. Modal shape functions

After solving for the natural frequencies, the corresponding modal shape can be determined. At eachnatural frequency, the matrix A in Eq. (17) is singular. Because of this, the constants Ci cannot be directlydetermined. However, a modal shape function can be obtained by setting one of the non-vanishingcoefficients, say, C1 equal to unity, and determining the others as a function of it.

In Figs. 6 and 7 shown are the second modal shape and in Figs. 8 and 9 the third modal shape of a simplysupported beam with two attached masses at locations: Z1 ¼ 0.25 and Z2 ¼ 0.5.

The modal shape of Figs. 6 and 8 were obtained by assuming the ratio ci ¼ 0.1 and varying the mass relationfor values Mi ¼ 0, 0.5, 1 and 2. In Figs. 7 and 9 the second and third modal shape, assuming Mi ¼ 1 andtaking different values for ci ¼ 0; 0.01; 0.05 and 0.1 are shown.

ARTICLE IN PRESS

Fig. 9. Third mode shape of a simply supported beam with two masses attached. —— ci ¼ 0; � � � � � � ci ¼ 0.01; – � – ci ¼ 0.05;

—�— ci ¼ 0.1 and Mi ¼ 1.


6. Conclusions

Usually when the effect of attached masses on vibrating beams is studied, only the translational inertia ofthe mass is considered. In those cases, it is observed in general, that natural frequencies decrease with respectto the values of the bare beam, except for the cases in which the masses are located at nodal points of thecorresponding normal mode (see the case of the fourth eigenvalue depicted in Tables 3 and 4, when therotatory inertia is not taken into account: c1 ¼ c2 ¼ 0).

On the other hand, when the model takes into account the rotatory inertia of the mass too, all the naturalfrequencies of vibration decrease. The influence of the rotatory inertia is larger on the upper frequencies. Thiseffect may be observed in Figs. 7 and 9. In Fig. 9, the variation of ci for the third modal shape has a largereffect than the one observed in Fig. 7 for the second mode shape.

The effect of the translatory inertia has its highest influence over a natural frequency, when the mass islocated at an antinode of the corresponding normal mode. In that situation the rotatory inertia has no effect(as it occurs with the second eigenvalue shown in Table 3).

The effect of the rotatory inertia has its highest influence when the mass is located at a node of the normalmode (Table 3 and 4, fourth eigenvalue).

Acknowledgments

The present work has been sponsored by Secretarıa General de Ciencia y Tecnologıa of UniversidadNacional del Sur at the Department of Engineering, and by CONICET Research and Development Program.

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Documents

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